The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the height is 10 yards and the base is 9 yards, we can substitute these values into the formula:
\[ A = \frac{1}{2} \times 9 \times 10 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 90 = 45 \]
So, the area of the triangle is \( 45 \) square yards.